Lectures: Tuesday 14:00-16:00, HG E 21 (Spring 2022 at ETH)

Course description: This course will cover various aspects of moduli space of stable bundles on curves, which is one of the most studied object in algebraic geometry. In the first part of the lectures, we will study basics about stability and go over the construction of the moduli space. In the second part, we will study some of the local (smoothness and dimension) and global (cohomology group) properties of the moduli space. In the last part, we will study another global invariants, called Verlinde numbers, which are governed by a fascinating formula first predicted from physics. 

Prerequisites: I will assume basics in algebraic geometry and algebraic topology. However, I will try to be as self-contained as possible. 

Grading: By a session examination (written, 120 minutes, open book).

Lecture note -- link

References: There is no definitive textbook for the lecture. However, you may find references below helpful. 

• J. Le Potier "Lectures on vector bundles", 1997.

• S. Mukai "An introduction to invariants and moduli", 2003.

• D. Huybrechts, M. Lehn "The geometry of moduli spaces of sheaves, second edition", 2010.

• D. Mumford, J. Fogarty, F. Kirwan "Geometric invariant theory", 1994.

• M. Thaddeus "Stable pairs, linear systems and the Verlinde formula", 1994.

Lecture plan: 

• Lecture 1: Introduction to the course with Grassmannian; construction, local and global properties, enumerative question. 

• Lecture 2: Coherent sheaf, sheaf cohomology, Chern classes, Hirzebruch-Riemann-Roch, Serre duality.

• Lecture 3: Classification problem for coherent sheaves and some difficulties. 

• Lecture 4: (Semi)stability, Harder-Narasimhan filtration, openness and boundedness of semistability. 

• Lecture 5: Introduction to geometric invariant theory and Hilbert-Mumford numerical criterion.

• Lecture 6: Moduli functor, corepresentability, construction of moduli space via GIT.

• Lecture 7: GIT construction continued.

• Lecture 8: Deformation theory of sheaves and local property of moduli spaces. 

• Lecture 9: No lecture for holiday.

• Lecture 10: Deformation theory continued. Geometric properties of moduli spaces (non-emptiness, unirationality).

• Lecture 11: Tautological classes, generators of the cohomology of moduli space, some relations between generators. 

• Lecture 12: Determinant line bundles and their properties.

• Lecture 13: Introduction to the Thaddeus' stable pairs.

• Lecture 14: Wall-crossing phenomenon of moduli space of stable pairs. 

• Lecture 15: Ample cone of moduli spaces and proof of Verline formula for rank 2 and odd degree.